In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R<sup>2n</sup>, ω（?）σ). Let us define l<sub>1</sub>（M, ω）=inf{【ω, α】|】...In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R<sup>2n</sup>, ω（?）σ). Let us define l<sub>1</sub>（M, ω）=inf{【ω, α】|】0, α∈π<sub>2</sub>（M）}. Suppose l<sub>1</sub>（M, ω）】O, O【πr<sup>2</sup>【2/1 l<sub>1</sub>（M, ω）. Then C<sub>HZ</sub>（M×B（r））=C<sub>HZ</sub>（M×Z（r））=πr<sup>2</sup>. In the case M is a point {P}, we obtain the well-known result at present. For n】1, consider on Cp<sup>n-1</sup> the standard symplectic form co such that ω[u]=n for a generator u of H<sub>2</sub>(CP<sup>n-1</sup>. Suppose O【πr<sup>2</sup>【2/1 n. ThenC<sub>HZ</sub>（M×B（r））=C<sub>HZ</sub>（M×Z（r））=πr<sup>2</sup>.As an application, we claim that the Weinstein conjecture in M×Z（r） is proved correct.展开更多
In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved...In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C-1-smooth hypersurface of contact type.展开更多
基金Project supported by the Science Foundation of Tsinghua University
文摘In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R<sup>2n</sup>, ω（?）σ). Let us define l<sub>1</sub>（M, ω）=inf{【ω, α】|】0, α∈π<sub>2</sub>（M）}. Suppose l<sub>1</sub>（M, ω）】O, O【πr<sup>2</sup>【2/1 l<sub>1</sub>（M, ω）. Then C<sub>HZ</sub>（M×B（r））=C<sub>HZ</sub>（M×Z（r））=πr<sup>2</sup>. In the case M is a point {P}, we obtain the well-known result at present. For n】1, consider on Cp<sup>n-1</sup> the standard symplectic form co such that ω[u]=n for a generator u of H<sub>2</sub>(CP<sup>n-1</sup>. Suppose O【πr<sup>2</sup>【2/1 n. ThenC<sub>HZ</sub>（M×B（r））=C<sub>HZ</sub>（M×Z（r））=πr<sup>2</sup>.As an application, we claim that the Weinstein conjecture in M×Z（r） is proved correct.
基金Supported by the NNSF of China(19971045) the MCF of Chinese University
文摘In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C-1-smooth hypersurface of contact type.
